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G R A D U A T E R E C O R D E X A M I N A T I O N S Mathematics TestPractice BookThis practice book contains䡲 one actual, full-length GRE Mathematics Test䡲 test-taking strategiesBecome familiar with䡲 test structure and content䡲 test instructions and answering proceduresCompare your practice test results with the performance of those whotook the test at a GRE administration.This book is provided FREE with test registration by the Graduate Record Examinations Board.www.ets.org/gre
Note to Test Takers: Keep this practice book until you receive your score report.This book contains important information about scoring. Copyright 2008 by Educational Testing Service. All rights reserved.ETS, the ETS logos, LISTENING. LEARNING. LEADING., GRADUATE RECORD EXAMINATIONS,and GRE are registered trademarks of Educational Testing Service (ETS) in the United States of Americaand other countries throughout the world.
Table of ContentsPurpose of the GRE Subject Tests . 3Development of the Subject Tests. 3Content of the Mathematics Test . 4Preparing for a Subject Test. 5Test-Taking Strategies . 5What Your Scores Mean . 6The GRE Board recommends that scores on theSubject Tests be considered in conjunction with otherrelevant information about applicants. Because numerous factors influence success in graduate school,reliance on a single measure to predict success is notadvisable. Other indicators of competence typicallyinclude undergraduate transcripts showing coursestaken and grades earned, letters of recommendation,and GRE General Test scores. For information aboutthe appropriate use of GRE scores, see the GRE Guideto the Use of Scores at ets.org/gre/stupubs.Practice Mathematics Test . 9Scoring Your Subject Test . 65Evaluating Your Performance . 68Answer Sheet. 69Purpose of theGRE Subject TestsThe GRE Subject Tests are designed to help graduateschool admission committees and fellowship sponsorsassess the qualifications of applicants in specific fieldsof study. The tests also provide you with an assessmentof your own qualifications.Scores on the tests are intended to indicateknowledge of the subject matter emphasized in manyundergraduate programs as preparation for graduatestudy. Because past achievement is usually a goodindicator of future performance, the scores are helpfulin predicting success in graduate study. Because the testsare standardized, the test scores permit comparisonof students from different institutions with differentundergraduate programs. For some Subject Tests,subscores are provided in addition to the total score;these subscores indicate the strengths and weaknessesof your preparation, and they may help you plan futurestudies.Development of theSubject TestsEach new edition of a Subject Test is developed bya committee of examiners composed of professors inthe subject who are on undergraduate and graduatefaculties in different types of institutions and indifferent regions of the United States and Canada.In selecting members for each committee, theGRE Program seeks the advice of the appropriateprofessional associations in the subject.The content and scope of each test are specifiedand reviewed periodically by the committee ofexaminers. Test questions are written by committeemembers and by other university faculty memberswho are subject-matter specialists. All questionsproposed for the test are reviewed and revised by thecommittee and subject-matter specialists at ETS. Thetests are assembled in accordance with the contentspecifications developed by the committee to ensureadequate coverage of the various aspects of the fieldand, at the same time, to prevent overemphasis onany single topic. The entire test is then reviewed andapproved by the committee.MATHEMATICS TESTPRACTICE BOOK3
Subject-matter and measurement specialists on theETS staff assist the committee, providing informationand advice about methods of test construction andhelping to prepare the questions and assemble the test.In addition, each test question is reviewed to eliminatelanguage, symbols, or content considered potentiallyoffensive, inappropriate for major subgroups of the testtaking population, or likely to perpetuate any negativeattitude that may be conveyed to these subgroups.Because of the diversity of undergraduate curricula,it is not possible for a single test to cover all the materialyou may have studied. The examiners, therefore, selectquestions that test the basic knowledge and skillsmost important for successful graduate study in theparticular field. The committee keeps the test up-todate by regularly developing new editions and revisingexisting editions. In this way, the test content remainscurrent. In addition, curriculum surveys are conductedperiodically to ensure that the content of a test reflectswhat is currently being taught in the undergraduatecurriculum.After a new edition of a Subject Test is firstadministered, examinees’ responses to each testquestion are analyzed in a variety of ways to determinewhether each question functioned as expected. Theseanalyses may reveal that a question is ambiguous,requires knowledge beyond the scope of the test, oris inappropriate for the total group or a particularsubgroup of examinees taking the test. Such questionsare not used in computing scores.Following this analysis, the new test edition isequated to an existing test edition. In the equatingprocess, statistical methods are used to assess thedifficulty of the new test. Then scores are adjusted sothat examinees who took a more difficult edition ofthe test are not penalized, and examinees who tookan easier edition of the test do not have an advantage.Variations in the number of questions in the differenteditions of the test are also taken into account in thisprocess.Scores on the Subject Tests are reported as threedigit scaled scores with the third digit always zero.The maximum possible range for all Subject Test totalscores is from 200 to 990. The actual range of scoresfor a particular Subject Test, however, may be smaller.For Subject Tests that report subscores, the maximumpossible range is 20 to 99; however, the actual range of4subscores for any test or test edition may be smaller.Subject Test score interpretive information is providedin Interpreting Your GRE Scores, which you will receivewith your GRE score report. This publication is alsoavailable at ets.org/gre/stupubs.Content of theMathematics TestThe test consists of approximately 66 multiple-choicequestions drawn from courses commonly offered atthe undergraduate level. Approximately 50 percent ofthe questions involve calculus and its applications—subject matter that can be assumed to be common tothe backgrounds of almost all mathematics majors.About 25 percent of the questions in the test are inelementary algebra, linear algebra, abstract algebra,and number theory. The remaining questions dealwith other areas of mathematics currently studied byundergraduates in many institutions.The following content descriptions may assiststudents in preparing for the test. The percents givenare estimates; actual percents will vary somewhat fromone edition of the test to another.Calculus—50%䡲 Material learned in the usual sequence ofelementary calculus courses—differentialand integral calculus of one and of severalvariables—includes calculus-based applicationsand connections with coordinate geometry,trigonometry, differential equations, and otherbranches of mathematicsAlgebra—25%䡲 Elementary algebra: basic algebraic techniquesand manipulations acquired in high school andused throughout mathematics䡲 Linear algebra: matrix algebra, systems of linearequations, vector spaces, linear transformations,characteristic polynomials, and eigenvalues andeigenvectors䡲 Abstract algebra and number theory: elementarytopics from group theory, theory of rings andmodules, field theory, and number theoryMATHEMATICS TESTPRACTICE BOOK
Additional Topics—25%䡲 Introductory real analysis: sequences andseries of numbers and functions, continuity,differentiability and integrability, and elementarytopology of and n䡲 Discrete mathematics: logic, set theory,combinatorics, graph theory, and algorithms䡲 Other topics: general topology, geometry,complex variables, probability and statistics, andnumerical analysisThe above descriptions of topics covered in the testshould not be considered exhaustive; it is necessary tounderstand many other related concepts. Prospectivetest takers should be aware that questions requiring nomore than a good precalculus background may be quitechallenging; such questions can be among the mostdifficult questions on the test. In general, the questionsare intended not only to test recall of information butalso to assess test takers’ understanding of fundamentalconcepts and the ability to apply those concepts invarious situations.Preparing for a Subject TestGRE Subject Test questions are designed to measureskills and knowledge gained over a long period of time.Although you might increase your scores to some extentthrough preparation a few weeks or months before youtake the test, last minute cramming is unlikely to be offurther help. The following information may be helpful.䡲 A general review of your college courses isprobably the best preparation for the test.However, the test covers a broad range of subjectmatter, and no one is expected to be familiarwith the content of every question.䡲 Use this practice book to become familiar withthe types of questions in the GRE MathematicsTest, taking note of the directions. If youunderstand the directions before you take thetest, you will have more time during the test tofocus on the questions themselves.Test-Taking StrategiesThe questions in the practice test in this bookillustrate the types of multiple-choice questions in thetest. When you take the actual test, you will mark youranswers on a separate machine-scorable answer sheet.Total testing time is two hours and fifty minutes; thereare no separately timed sections. Following are somegeneral test-taking strategies you may want to consider.䡲 Read the test directions carefully, and work asrapidly as you can without being careless. Foreach question, choose the best answer from theavailable options.䡲 All questions are of equal value; do not wastetime pondering individual questions you findextremely difficult or unfamiliar.䡲 You may want to work through the test quiterapidly, first answering only the questions aboutwhich you feel confident, then going back andanswering questions that require more thought,and concluding with the most difficult questionsif there is time.䡲 If you decide to change an answer, make sureyou completely erase it and fill in the ovalcorresponding to your desired answer.䡲 Questions for which you mark no answer or morethan one answer are not counted in scoring.䡲 Your score will be determined by subtractingone-fourth the number of incorrect answers fromthe number of correct answers. If you have someknowledge of a question and are able to rule outone or more of the answer choices as incorrect,your chances of selecting the correct answer areimproved, and answering such questions willlikely improve your score. It is unlikely that pureguessing will raise your score; it may lower yourscore.䡲 Record all answers on your answer sheet.Answers recorded in your test book will notbe counted.䡲 Do not wait until the last five minutes of a testingsession to record answers on your answer sheet.MATHEMATICS TESTPRACTICE BOOK5
What Your Scores MeanYour raw score — that is, the number of questions youanswered correctly minus one-fourth of the numberyou answered incorrectly — is converted to the scaledscore that is reported. This conversion ensures thata scaled score reported for any edition of a SubjectTest is comparable to the same scaled score earnedon any other edition of the same test. Thus, equalscaled scores on a particular Subject Test indicateessentially equal levels of performance regardless ofthe test edition taken. Test scores should be comparedonly with other scores on the same Subject Test. (Forexample, a 680 on the Computer Science Test is notequivalent to a 680 on the Mathematics Test.)Before taking the test, you may find it usefulto know approximately what raw scores would berequired to obtain a certain scaled score. Severalfactors influence the conversion of your raw scoreto your scaled score, such as the difficulty of the testedition and the number of test questions included inthe computation of your raw score. Based on recenteditions of the Mathematics Test, the following tablegives the range of raw scores associated with selectedscaled scores for three different test editions. (Notethat when the number of scored questions for a giventest is greater than the number of actual scaled scorepoints, it is likely that two or more raw scores willconvert to the same scaled score.) The three testeditions in the table that follows were selected toreflect varying degrees of difficulty. Examinees shouldnote that future test editions may be somewhat moreor less difficult than the test editions illustrated in thetable.6Range of Raw Scores* Neededto Earn Selected Scaled Score onThree Mathematics TestEditions that Differ in DifficultyRaw ScoresScaled ScoreForm AForm BForm C800494745700393635600282525500181416Number of Questions Used to Compute Raw Score666666*Raw Score Number of correct answers minus one-fourth thenumber of incorrect answers, rounded to the nearest integer.For a particular test edition, there are many ways toearn the same raw score. For example, on the editionlisted above as “Form A,” a raw score of 28 would earna scaled score of 600. Below are a few of the possibleways in which a scaled score of 600 could be earned onthe edition:Examples of Ways to Earna Scaled Score of 600 on theEdition Labeled as “Form MATHEMATICS TESTPRACTICE nswered38190Number ofQuestionsUsed toComputeRaw Score666666
PRACTICE TESTTo become familiar with how the administration will be conducted at the test center, first remove theanswer sheet (pages 69 and 70). Then go to the back cover of the test book (page 64) and follow theinstructions for completing the identification areas of the answer sheet. When you are ready to begin thetest, note the time and begin marking your answers on the answer sheet.MATHEMATICS TESTPRACTICE BOOK7
FORM GR056868GRADUATE RECORD EXAMINATIONS MATHEMATICS TESTDo not break the sealuntil you are told to do so.The contents of this test are confidential.Disclosure or reproduction of any portionof it is prohibited.THIS TEST BOOK MUST NOT BE TAKEN FROM THE ROOM.Copyright 1999, 2000, 2003, 2005 by Educational Testing Service. All rights reserved.GRE, GRADUATE RECORD EXAMINATIONS, ETS, EDUCATIONAL TESTINGSERVICE and the ETS logos are registered trademarks of Educational Testing Service.9
MATHEMATICS TESTTime—170 minutes66 QuestionsDirections: Each of the questions or incomplete statements below is followed by five suggested answers orcompletions. In each case, select the one that is the best of the choices offered and then mark the correspondingspace on the answer sheet.Computation and scratch work may be done in this examination book.Note: In this examination:(1) All logarithms with an unspecified base are natural logarithms, that is, with base e.(2) The set of all real numbers x such that a x b is denoted by a, b@.(3) The symbols , , , and denote the sets of integers, rational numbers, real numbers,and complex numbers, respectively.1. In the xy-plane, the curve with parametric equations x(B) p(A) 3(C) 3p(D)32cos t and y(E)sin t , 0x(B) yx 1(C) yx 2(D) y2x(E) y2x 1Unauthorized copying or reuse ofany part of this page is illegal.10p , has lengthp22. Which of the following is an equation of the line tangent to the graph of y(A) ytx e x at x0?GO ON TO THE NEXT PAGE.
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3. If V and W are 2-dimensional subspaces of 4 , what are the possible dimensions of the subspace V W ?(A) 1 only(B) 2 only(C) 0 and 1 only(D) 0, 1, and 2 only(E) 0, 1, 2, 3, and 44. Let k be the number of real solutions of the equation e x x 2 0 in the interval 0, 1@, and let n be thenumber of real solutions that are not in 0, 1@. Which of the following is true?(A) k0 and n1Unauthorized copying or reuse ofany part of this page is illegal.12(B) k1 and n0(C) kn1(D) k ! 1(E) n ! 1GO ON TO THE NEXT PAGE.
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5. Suppose b is a real number and f xgraphed above. Then f 5(A) 15(B) 27(C) 673 x 2 bx 12 defines a function on the real line, part of which is(D) 72(E) 876. Which of the following circles has the greatest number of points of intersection with the parabola x2(A) x2 y 21(B) x2 y 22(C) x2 y 29(D) x2 y 216(E) x2 y 225Unauthorized copying or reuse ofany part of this page is illegal.14y 4?GO ON TO THE NEXT PAGE.
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3Ô 3 x 1 dx7.(A) 0(B) 5(C) 10(D) 15(E) 208. What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 andthe other two vertices on the circle?(A)12(B) 1(C)2(D) pJKL(E)1 241Ô01 x 4 dx1Ô01 x 4 dx11 x8 dxÔ09. Which of the following is true for the definite integrals shown above?(A) J L 1 K(B) J L K 1(C) L J 1 K(D) L J K 1(E) L 1 J KUnauthorized copying or reuse ofany part of this page is illegal.16GO ON TO THE NEXT PAGE.
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10. Let g be a function whose derivative g is continuous and has the graph shown above. Which of the followingvalues of g is largest?(A) g 1(B) g 2(C) g 3(D) g 411. Of the following, which is the best approximation of 1.5 266(A) 1,000(B) 2,700(C) 3,200(D) 4,100(E) g 532?(E) 5,30012. Let A be a 2 2 matrix for which there is a constant k such that the sum of the entries in each row and eachcolumn is k. Which of the following must be an eigenvector of A ?I.10II.01III.11(A) I only(B) II onlyUnauthorized copying or reuse ofany part of this page is illegal.18(C) III only(D) I and II only(E) I, II, and IIIGO ON TO THE NEXT PAGE.
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13. A total of x feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible areaof the yard, in terms of x ?(A)x29(B)x28(C)x24(D) x 2(E) 2x 214. What is the units digit in the standard decimal expansion of the number 725 ?(A) 1(B) 3(C) 5(D) 7(E) 915. Let f be a continuous real-valued function defined on the closed interval 2, 3@. Which of the following isNOT necessarily true?(A) f is bounded.(B)Ô3 2f t dt exists.(C) For each c between f 2 and f 3 , there is an x 2, 3@ such that f x(D) There is an M in f(E) limh 0Ô3 2f t dt5M .f h f 0exists.hUnauthorized copying or reuse ofany part of this page is illegal.20 2, 3@ such thatc.GO ON TO THE NEXT PAGE.
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16. What is the volume of the solid formed by revolving about the x-axis the region in the first quadrant of the1xy-plane bounded by the coordinate axes and the graph of the equation y?1 x2(A)p2(B) p(C)p24(D)p22(E) 17. How many real roots does the polynomial 2 x 5 8 x 7 have?(A) None(B) One(C) Two(D) Three(E) Five18. Let V be the real vector space of all real 2 3 matrices, and let W be the real vector space of all real 4 1column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace v V : T v 0 ?(A) 2(B) 3Unauthorized copying or reuse ofany part of this page is illegal.22(C) 4(D) 5(E) 6GO ON TO THE NEXT PAGE.
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19. Let f and g be twice-differentiable real-valued functions defined on . If fof the following inequalities must be true for all x 0 ?(A) f x(B) fxg x for all x0, whichg xxgx(C) f xf 0(D) fxf 0(E) fxfg x0g 0g xgg 0xg020. Let f be the function defined on the real line byf xx2x3if x is rationalif x is irrational.If D is the set of points of discontinuity of f, then D is the(A) empty set(B) set of rational numbers(C) set of irrational numbers(D) set of nonzero real numbers(E) set of real numbers21. Let P1 be the set of all primes, 2, 3, 5, 7, . . . , and for each integer n, let Pn be the set of all prime multiplesof n, 2 n, 3n, 5n, 7n, . . . . Which of the following intersections is nonempty?(A) P1P23Unauthorized copying or reuse ofany part of this page is illegal.24(B) P7P21(C) P12P20(D) P20P24(E) P5P25GO ON TO THE NEXT PAGE.
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22. Let C be the collection of all continuous functions from to . Then C is a real vector space withpointwise addition and scalar multiplication defined byf g xf x g x and rfxrf xfor all f , g C and all r, x . Which of the following are subspaces of C ?I. f : f is twice differentiable and f x 2 f x 3 f xII. g : g is twice differentiable and g x3g xIII. h : h is twice differentiable and h x(A) I only(B) I and II only23. For what value of b is the line y(A)10e(B) 10(B) e2Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.26for all x h x 1 for all x (C) I and III only(D) II and III only10 x tangent to the curve y(D) e10(C) 10e24. Let h be the function defined by h x(A) e 10 for all x x2 x tÔ0(C) e2 ee(E) I, II, and IIIebx at some point in the xy-plane?(E) edt for all real numbers x. Then h 1(D) 2e2(E) 3e2 eGO ON TO THE NEXT PAGE.-18-
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25. Let an n1(A) 15 3126. Let f x, ybe defined recursively by a1(B) 30 311 and an 1(C)3129n 2an for n 1. Then a30 is equal ton(D)3230(E)32!30! 2!x 2 2 xy y3 for all real x and y. Which of the following is true?(A) f has all of its relative extrema on the line xy.(B) f has all of its relative extrema on the parabola xy2 .(C) f has a relative minimum at 0, 0 .(D) f has an absolute minimum at2 2, .3 3(E) f has an absolute minimum at 1, 1 .Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.28GO ON TO THE NEXT PAGE.-20-
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27. Consider the two planes x 3 y 2 zintersection of these planes?7 and 2 x y 3z0 in 3 . Which of the following sets is the(A) (B) 0, 3, 1 (C) x, y, z : xt, y(D) x, y, z : x7t , y(E) x, y, z : x 2 y z7 2t , t 3t, z3 t, z1 5t , t 7 28. The figure above shows an undirected graph with six vertices. Enough edges are to be deleted from the graphin order to leave a spanning tree, which is a connected subgraph having the same six vertices and no cycles.How many edges must be deleted?(A) One(B) TwoUnauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.30(C) Three(D) Four(E) FiveGO ON TO THE NEXT PAGE.-22-
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29. For all positive functions f and g of the real variable x, letf苲g(B)f苲gf xg xif and only if limxWhich of the following is NOT a consequence of f(A) f 2 苲 g 2苲 be a relation defined by(C) e f1.苲g?苲 eg(D) fg苲 2g(E) g 苲 f30. Let f be a function from a set X to a set Y. Consider the following statements.P: For each x X , there exists y Y such that f xQ: For each y Y , there exists x X such that f xR: There exist x1, x2 X such that x1 x2 and f x1y.y.f x2 .The negation of the statement “ f is one-to-one and onto Y ” is(A) P or not R(B) R or not P(C) R or not Q(D) P and not R(E) R and not QUnauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.32GO ON TO THE NEXT PAGE.-24-
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31. Which of the following most closely represents the graph of a solution to the differential equation(A)(B)(D)(E)Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.34dydx1 y4 ?(C)GO ON TO THE NEXT PAGE.-26-
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and , are defined on a nonempty set S, and that the32. Suppose that two binary operations, denoted byfollowing conditions are satisfied for all x, y, and z in S:y and x y are in S.y zx yz and xy y x(1) x(2) x(3) xyzxyz.Also, for each x in S and for each positive integer n, the elements nx and x n are defined recursively asfollows:1xx1x andif kx and x k have been defined, then k1 xkxx and x k1xkx.Which of the following must be true?I. xII. n xIII. x m(A) I onlyynyxnxny n for all x and y in S and for each positive integer n.nxny for all x and y in S and for each positive integer n.xmn(B) II onlyUnauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.36for each x in S and for all positive integers m and n.(C) III only(D) II and III only(E) I, II, and IIIGO ON TO THE NEXT PAGE.-28-
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33. The Euclidean algorithm is used to find the greatest common divisor (gcd) of two positive integers a and b .input(a)input(b)while b 0beginr : a mod ba : bb : rendgcd : aoutput(gcd)When the algorithm is used to find the greatest common divisor of a 273 and b 110, which of thefollowing is the sequence of computed values for r ?(A) 2, 26, 1, 0(B) 2, 53, 1, 0(C) 53, 2, 1, 0(D) 53, 4, 1, 0(E) 53, 5, 1, 034. The minimal distance between any point on the sphere x 2sphere x 3(A) 02 y 2(B) 4Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.382 z 4(C)2722 y 12 z 321 and any point on the4 is(D) 2 2 1(E) 3 3 1GO ON TO THE NEXT PAGE.-30-
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35. At a banquet, 9 women and 6 men are to be seated in a row of 15 chairs. If the entire seating arrangement is to bechosen at random, what is the probability that all of the men will be seated next to each other in 6 consecutivepositions?(A)1È15ØÉÊ 6 ÙÚ(B)6!È15ØÉÊ 6 ÙÚ(C)10!15!(D)6! 9!14!(E)6!10!15!36. Let M be a 5 5 real matrix. Exactly four of the following five conditions on M are equivalent to each other.Which of the five conditions is equivalent to NONE of the other four?(A) For any two distinct column vectors u and v of M, the set u, v is linearly independent.(B) The homogeneous system M x(C) The system of equations M x0 has only the trivial solution.b has a unique solution for each real 5 1 column vector b.(D) The determinant of M is nonzero.(E) There exists a 5 5 real matrix N such that NM is the 5 5 identity matrix.37. In the complex z-plane, the set of points satisfying the equation z 2z 2 is a(A) pair of points(B) circle(C) half-line(D) line(E) union of infinitely many different linesUnauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.40GO ON TO THE NEXT PAGE.-32-
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38. Let A and B be nonempty subsets of and let f : Afollowing must be true?(A) C1f(B) Dff1B, which of theD1f CC(D) f1f Cf1ffA and Df C(C) f(E) fB be a function. If CDf11DD39. In the figure above, as r and s increase, the length of the third side of the triangle remains 1 and the measure ofthe obtuse angle remains 110 . What is lim s r ?sr(A) 0(B) A positive number less than 1(C) 1(D) A finite number greater than 1(E) Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.42GO ON TO THE NEXT PAGE.-34-
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40. For which of the following rings is it possible for the product of two nonzero elements to be zero?(A) The ring of complex numbers(B) The ring of integers modulo 11(C) The ring of continuous real-valued functions on 0, 1@(D) The ring a b 2 : a and b are rational numbers (E) The ring of polynomials in x with real coefficients41. Let C be the circle x2 y 2vÔ C1 oriented counterclockwise in the xy-plane. What is the value of the line integral2 x y dx x 3 y dy ?(A) 0(B) 1(C)p2(D) p(E) 2 p42. Suppose X is a discrete random variable on the set of positive integers such that for each positive integer n, the1probability that X n is n . If Y is a random variable with the same probability distribution and X and Y2are independent, what is the probability that the value of at least one of the variables X and Y is greater than 3 ?(A)164(B)1564Unauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.44(C)14(D)38(E)49GO ON TO THE NEXT PAGE.-36-
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43. If ze2 p i 5 , then 1 z z 2 z3 5z 4 4 z 5 4 z 6 4 z 7 4 z8 5z 9(B) 4e3p i(A) 05(C) 5e 4 p i5(D) 4e2 p i5(E) 5e3 p i544. A fair coin is to be tossed 100 times, with each toss resulting in a head or a tail. If H is the total number of headsand T is the total number of tails, which of the following events has the greatest probability?(A) H50(B) T 60(C) 51H55(D) H 48 and T 48(E) H5 or H 9545. A circular region is divided by 5 radii into sectors as shown above. Twenty-one points are chosen in the circularregion, none of which is on any of the 5 radii. Which of the following statements must be true?I. Some sector contains at least 5 of the points.II. Some sector contains at most 3 of the points.III. Some pair of adjacent sectors contains a total of at least 9 of the points.(A) I only(B) III onlyUnauthorized copying or reuse ofany part of this page is illegal.Unauthorized copying or reuse ofany part of this page is illegal.46(C) I and II only(D) I and III only(E) I, II, and IIIGO ON TO THE NEXT PAGE.-38-
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46. Let G be the group of complex numbers 1, i, 1, i under multiplication. Which of the following statementsare true about the homomorphisms of G into itself?z defines one such homomorp
GRE Subject Tests The GRE Subject Tests are designed to help graduate school admission committees and fellowship sponsors assess the qualifi cations of applicants in specifi c fi elds of study. The tests also provide you with an assessment of your own qualifi cations. Scores on the tests are intended
